In communication systems a modem is used to convert (modulate) digital signals generated by a computer into analog signals suitable for transmission over telephone lines. Another modem, located at the receiving end of the transmission, converts (demodulates) the analog signals back into digital form. In a particular modulation transmission scheme, the phase and amplitude of a signal are shifted to various combinations of values, each combination indicating a different set of transmitted bits. At the receiver, proper decoding includes detecting the various phase and amplitude combinations. In a two dimensional modulation scheme, the signal can be represented mathematically with an I (in-phase) component and a Q (quadrature-phase) component of the signal, each of which is .pi./20 out of phase with respect to the other. The plot of these two components on a two dimensional graph for a set of received symbols results in a pattern referred to as a constellation.
Proper detection of the I and Q components of the signal is hampered by various sources of signal degradation. One such source is intersymbol interference where consecutive transmitted symbols interfere with each other. Other sources of signal degradation include the transmission media (i.e. wire) and analog filters. These factors produce large amplitude and group delay distortion in the signal that needs compensation.
To compensate for intersymbol interference (ISI) and other sources of signal degradation and distortion, best performance is achieved by implementing an equalizer as a fractionally spaced adaptive filter. An adaptive filter can modify from time instant to time instant, the coefficients, also referred to as tap weights, used in the filter to remove ISI and to compensate for amplitude and group delay distortions. The update of the tap weights is done to minimize the error at the output of the filter. This error is effectively a measure of the difference between the actual output of the filter and the expected output. The adaptive process continues until the error is at a minimum (i.e. the filter converges).
The convergence of an equalizer depends on many factors including initial tap weights, desired convergence rate, signal to noise ratio (SNR) at the input and phase changes caused by a clock recovery circuit at the receiver, and can be accomplished with various adaptive algorithms.
The adaptation of the tap weights in adaptive equalizers is based on an assumed correct decision about which symbol was received. This assumption is valid for equalizers with a training sequence for which the received symbol is in fact known in advance. Equalizers, however, are also used without the benefit of a training sequence, in which case the decision is not necessarily correct. These equalizers are referred to as blind equalizers. The term blind refers to trying to find the correct equalizer coefficients without a reference training sequence, therefore during convergence the decisions may be incorrect and the coefficients (weights) erroneously updated. Although the possibility of a mistake exists, if the blind equalizer makes correct decisions for a sufficiently large set of received symbols, the equalizer will converge correctly.
A common problem with equalization is the detection of misconvergence to ensure that the convergence algorithm has worked properly. In certain cases of pathological misconvergence (discussed in detail below) typical equalizer error measures used to detect misconvergence (such as average error from filter outputs) will be below acceptable thresholds yet the constellation map for a given QAM will not be formed correctly (i.e. in number of points and/or size).